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The 1/N investment strategy is optimal under high modelambiguity
Georg Ch. Pfluga, Alois Pichlera, David Wozabalb,∗
aInstitute of Statistics and Operations Research, University of Vienna, Universitatsstraße 5/9, 1010, AustriabInstitute of Business Administration, University of Vienna, Brunner Straße 72, 1210, Austria
Abstract
The 1/N investment strategy, i.e. the strategy to split one’s wealth uniformly between the
available investment possibilities, recently received plenty of attention in the literature. In
this paper, we demonstrate that the uniform investment strategy is rational in situations where
an agent is faced with a sufficiently high degree of model uncertainty in the form of ambiguous
loss distributions. More specifically, we use a classical risk minimization framework to show
that, for a broad class of risk measures, as the uncertainty concerning the probabilistic model
increases, the optimal decisions tend to the uniform investment strategy.
To illustrate the theoretical results of the paper, we investigate the Markowitz portfolio
selection model as well as Conditional Value-at-Risk minimization with ambiguous loss distri-
butions. Subsequently, we set up a numerical study using real market data to demonstrate the
convergence of optimal portfolio decisions to the uniform investment strategy.
JEL classification: C44; D14; D81; G11
Keywords: Model uncertainty; Risk aware planning; Robust optimization
1. Introduction
The uniform investment strategy is interesting for researchers as well as practitioners for
two reasons. Firstly, comparative studies show that naive diversification is hard to outperform
as an investment strategy in a portfolio management context. Secondly, behavioral studies show
∗Corresponding author. Tel.: +43 1 4277 38105; fax: +43 1 4277 38104.Email addresses: [email protected] (Georg Ch. Pflug), [email protected]
(Alois Pichler), [email protected] (David Wozabal)
Preprint submitted to Banking & Finance July 25, 2011
that it is applied by agents in many situations. This is explained in the literature either by an
inherent psychological bias, leading to potentially irrational decisions, or by the presence of
some fundamental uncertainty in the decision model of the agent, making uniform diversifica-
tion a rational strategy to follow. The contribution of this paper falls into the latter category
as we argue that uniform diversification is an optimal strategy for certain types of risk averse
investors facing model uncertainty in a stochastic programming context.
The authors do not want to imply that uniform diversification is a recommendable invest-
ment strategy in general. However, based on the results of the paper, one can explain the relative
success of the 1/N rule in a stochastic portfolio optimization context as the result of an inac-
curate specification of the data generating process, i.e. a lack of accuracy in the modeling of
the distributions of the random asset returns. If the true model remains sufficiently ambiguous,
uniform diversification may outperform more sophisticated approaches.
We start our exposition by a literature review.
The uniform investment strategy can be traced back to the 4th century, when Rabbi Issac
bar Aha gave the following advice: ”One should always divide his wealth into three parts: a
third in land, a third in merchandise, and a third ready to hand”. 1
Of course, an asset allocation strategy as simple as the rule to divide the available capital
evenly among some (or even all) investment opportunities falls short of the sophistication of
modern portfolio theory, which in broad terms states that a portfolio should strike an optimal
balance between the prospective return of an investment and the possible risks of investing.
The optimal decision depends on the risk preferences of the investor. It can be seen as an irony
that Markowitz, arguably the father of modern portfolio theory, answered the question how he
manages his own funds by stating: ”My intention was to minimize my future regret. So I split
my contributions fifty-fifty between bonds and equities.” (see Zweig, 1998) – an application of
the 1/N rule on an aggregate level.
In a recent paper, DeMiguel et al. (2009b) use the 1/N strategy as a benchmark in a rolling
horizon setting and compare it against several portfolio optimization strategies. The models
1Babylonian Talmud: Tractate Baba Mezi’a, folio 42a
2
include the classical Markowitz portfolio selection rule as well as its most prominent exten-
sions like Bayesian-Shrinkage type estimators, aimed at dampening the effects of estimation
error, and more recent approaches based on the investors beliefs about several competing asset
pricing models. Furthermore, the authors include approaches that try to minimize the influence
of estimation errors by restricting the asset weights or entirely focussing on the risk minimal
portfolio (ignoring the expected loss dimension altogether). The results show that the bench-
mark 1/N rule outperforms most of the other more involved strategies in terms of Sharpe ratio,
certainty equivalent, and turnover and is not consistently outperformed by any of the models
considered in the study. The authors explain the results by stating that the errors in estima-
tion of the parameters of the optimization models outweigh the gains of the more advanced
methodology. Chan et al. (1999); Jagannathan and Ma (2003) conduct similar studies and also
conclude that it is hard to find an investment policy that consistently outperforms the uniform
investment strategy. Several authors try to incorporate this finding in their proposed portfolio
selection framework, see for example DeMiguel et al. (2009a); Tu and Zhou (2011).
Apart from the success of the 1/N rule in empirical studies, there is evidence that uniform
investment strategies are actually used in a multitude of situations where agents have to de-
cide on a mix of different alternatives. Benartzi and Thaler (2001) conduct experiments, where
subjects are asked to allocate money to different funds available in hypothetical defined con-
tribution pension plans. The authors find that a significant share of the investors use the 1/N
rule. This choice seems to be independent of the variety of funds offered, i.e. subjects that were
offered more equity funds invested more money in equity than subjects that were confronted
with an asset universe consisting of relatively fewer equity funds and more bonds. This leads
the authors to the conclusion that there is a natural psychological bias towards the 1/N strategy,
which may result in clearly irrational and even contradictory decisions. This can be interpreted
as a cognitive bias in the sense of Tversky and Kahneman (1981); Kahneman (2003). In Hu-
berman and Jiang (2006), a paper motivated by the work of Benartzi and Thaler (2001), data
on the choice of consumers in actual 401(k) plans is analyzed. The authors find that there is
a significant share of investors (roughly two thirds) that follow the uniform investment rule.
However, there is no statistical evidence of irrational behavior of the type found in the experi-
3
mental studies by Benartzi and Thaler (2001).
Other studies investigating the same phenomena in different situations, under the name of
diversification heuristic, diversification bias, or variety seeking, arrive at similar conclusions.
Simonson (1990) observes variety seeking behavior in setups where multiple decisions on fu-
ture consumption have to be taken as opposed to sequential decisions on immediate consump-
tion. In Simonson and Winer (1992), an analysis of yoghurt purchases of families reveals that
larger purchases (representing simultaneous decisions on future consumption) are significantly
more diverse than purchases of smaller quantities by the same families. The larger purchases
contain varieties which are otherwise not bought at all. The authors explain their findings by
rational risk minimizing behavior of the subjects facing uncertain future preferences. On the
contrary, Read and Loewenstein (1995) explain variety seeking behavior in simultaneous deci-
sions for future consumption by cognitive deficits termed time contraction and choice bracket-
ing. The former refers to a situation where the consumer underestimates the time between the
consumption of goods and thereby overestimates the satiation effect resulting from consuming
the same product, while the latter describes the phenomena that simultaneous choices are often
framed as a single portfolio choice encouraging diversification.
As mentioned before, the explanations offered in the literature for the empirical prevalence
of 1/N heuristics can be divided into papers conjecturing that there are inherent psychological
patterns which encourage the use of uniform investment decisions, even in situations where it
is disadvantageous, and approaches which try to find a rationale for this behavior. The latter
usually refers to some kind of fundamental uncertainty about the optimization problem involved
in the decision situation, making simple uniform diversification a rational strategy to follow.
The contribution of this paper is to show that this is indeed the case in portfolio optimization
problems under uncertainty if the distribution of the returns is ambiguous.
We consider a rational investor who tries to minimize her risk by choosing a portfolio of
assets with uncertain returns. While the investor has some prior information about the possible
distributions of the asset returns, the distribution is not exactly known. Hence, additional to
the uncertainty about the return, there is another layer of model uncertainty present, which we
will call ambiguity (also called epistemic or Knightian uncertainty after Frank Knight). Note
4
that, this kind of uncertainty is similar to the uncertainty used as justification of the 1/N rule in
Simonson (1990) and DeMiguel et al. (2009b) as it involves uncertainty about the nature of the
optimization problem faced by the decision maker.
The investor deals with this uncertainty by adopting a worst case approach and minimizing
the worst case risk under all distributions which seem plausible given the available information.
In accordance with the terminology in Ben-Tal et al. (2009), we call this set of distributions the
ambiguity set. We construct ambiguity sets as non-parametric neighborhoods of the prior in a
way which is natural from a mathematical statistics’ viewpoint. Subsequently, we show that
under weak conditions on the risk preferences of the investor, the optimal decisions approach
portfolios which obey the 1/N rule as the amount of model uncertainty increases.
The idea of robustifying portfolio selection problems with respect to ambiguity about the
distribution of future returns is not new and is mostly pursued in the Operations Research lit-
erature. See Maenhout (2004); Calafiore (2007); Pflug and Wozabal (2007); Garlappi et al.
(2007); Quaranta and Zaffaroni (2008); Vrontos et al. (2008); Kerkhof et al. (2010); Lutgens
and Schotman (2010); Tarashev (2010); Wozabal (2010) for recent advances in this direction.
The proposed approaches differ in the way the ambiguity sets are defined and in the methods
applied to solve the resulting optimization problems. Most of the papers make strong assump-
tions on the nature of the ambiguity to be able to deal with the robustified problems. Other
papers that use non-parametric methods similar to our approach are Calafiore (2007), Pflug and
Wozabal (2007) and Wozabal (2010). A comprehensive summary is beyond the scope of this
paper.
The paper is organized as follows: in Section 2 we set up portfolio optimization problems
under ambiguity and discuss how to quantify the degree of model uncertainty by the use of
probability metrics. Furthermore, we discuss how the Markowitz functional as well as the
Conditional Value-at-Risk fit in this framework. Section 3 contains the main theoretical results
of the paper, which permit us to identify the uniform investment strategy as optimal strategy as
model uncertainty increases. In Section 4, we demonstrate the theoretical results in numerical
studies based on real market data. We study the ambiguous Markowitz portfolio selection
model as well as the Conditional Value-at-Risk in detail. Section 5 concludes the paper by
5
summarizing the findings as well as outlining the implications of the results.
2. Investing under ambiguity
We consider an asset universe of N financial assets with random future losses and analyze
the decision problem of an agent who wants to invest a fixed amount of money in a combination
of these assets for one period of time. We model the investment decision as relative, possibly
negative, weights w = (w1, . . . ,wN)⊤ ∈ RN assigned to the assets. The investor has beliefs
about the joint distribution of future losses, which we describe by a prior distribution on RN .
Let (Ω, σ, µ) be a fixed probability space which admits a random variable XP : (Ω, σ, µ) →
RN with image measure P for each Borel measure on RN (see Lemma 2 in the Appendix for
a justification of this assumption). This assumption permits us to use the terms distribution
and probability measure synonymously. We will denote by || · ||Lp the norm in Lp(Ω, σ, µ) to
distinguish it from the vector norm || · ||p in RN .
Assume that the risk preferences of the investor can be described by a risk functional
R : Lp(Ω, σ, µ) → R, which assigns a real value to random variables X : (Ω, σ, µ) → R,
representing random future losses. The risk functional quantifies the riskiness of X, i.e. higher
values indicate more risk and thereby less desirable random variables. There is a plethora of
risk functionals discussed in the literature. However, in this paper, we mostly concentrate on
the following two well known functionals:
1. The Markowitz functional
Mγ(X) = E(X) + γ√
Var(X), (1)
where E(X) is the expectation of X and Var is the variance, while the parameter γ > 0
represents the risk aversion of the decision maker.
2. The Conditional Value-at-Risk (also called Average Value-at-Risk)
CVaRα(X) =1
1 − α
∫ 1
α
F−1X (t)dt,
where FX is the cumulative distribution function of the random variable X, and F−1X de-
notes it’s inverse distribution function. Note that since we define the Conditional Value-
6
at-Risk as a risk functional, we are concerned with the values in the upper tail of the loss
distribution, i.e. α will typically be chosen close to 1.
If the investor was sure that P is an accurate description of the future distribution of losses,
then she would decide on a portfolio composition w ∈ RN by solving the following single stage
stochastic programming problem
infw∈RN R(⟨XP,w⟩)
s.t. ⟨w,1⟩ = 1,(2)
where ⟨·, ·⟩ : RN × RN → R is the inner product, and 1 ∈ RN is a vector of ones. Note that
⟨XP,w⟩ = ∑Nn=1 wnXP
n : (Ω, σ, µ) → R, and the risk R(⟨XP,w⟩) depends on the probability
measure P on RN as well as the portfolio decision w ∈ RN . We will assume throughout the
paper that problem (2) is well-posed – in particular, we require that (2) is bounded from below.
However, in most real life situations the measure P is not known to the decision maker.
While statistical methods, analysis of fundamentals, and expert opinion can help to form a be-
lief about the measure P, the true distribution remains uncertain. It is, therefore, reasonable to
assume that the decision maker takes the available information into account, but also accounts
for model uncertainty in her decisions. As mentioned before, we model this uncertainty by
specifying a set of possible loss distributions given the prior information represented by a dis-
tribution P. This set of distributions is referred to as ambiguity set, and P is called the reference
probability measure. The ambiguity set consists of measures whose distance to the reference
measure does not exceed a certain threshold. To this end, we denote by P(RN) the space of all
Borel probability measures on RN , and by
d(·, ·) : P(RN) × P(RN)→ R+ ∪ 0
a metric on this space (see Gibbs and Su (2002) for a short introduction to the subject of
probability metrics). The ambiguity set can then be defined as
Bκ(P) = Q ∈ P : d(P,Q) ≤ κ ,
i.e. the ball of radius κ around the reference measure P.
7
In this paper, we focus on the Kantorovich or Wasserstein metric, i.e. we choose d(·, ·) as
dp(P,Q) = inf
(∫
RN×RN||x − y||ppdπ(x, y)
) 1p
: proj1(π) = P, proj2(π) = Q
, (3)
where the infimum runs over all transportation plans, viz. joint distributions π on RN × RN
and proj1(π), proj2(π) are the marginal distribution of the first N and the last N components
respectively. It is well known that the infimum in the above definition is always attained (see
Villani, 2003).
One reason for choosing the Kantorovich distance is that it plays an important role in sta-
bility results in stochastic programming, see for example Mirkov and Pflug (2007); Heitsch and
Romisch (2009).
Furthermore, the Kantorovich metric dp metrizises weak convergence on sets of probability
measures on RN for which x 7→ ∥x∥pp is uniformly integrable (see Villani, 2003). In particular,
the empirical measure Pm based on m observations, satisfies
dp(P, Pm)m→∞−→ 0,
if the p-th moment of P exists. This property justifies the use of dp to construct ambiguity sets:
a stronger metric would not necessarily allow to reduce the degree of ambiguity by collecting
more data, while a weaker metric would lead to a topology which permits too many convergent
sequences. A particularly interesting alternative would be the Kullback-Leibler distance, which
is used in Calafiore (2007) as well as in Kovacevic (2011) in a robust programming context.
Since this metric is stronger than the Kantorovich distance, the results of this paper do not
ensure that the optimal portfolio for a high level of ambiguity is the uniform portfolio.
Since dp is closely related to the concept of weak convergence, there exist a range of finite
sample results making it possible to interpret Kantorovich balls as confidence sets around the
empirical measure. See for example Dudley (1968) for completely nonparametric bounds, or
Kersting (1978) for bounds which require certain smoothness properties of the true measure.
Ideas on how to use these results to construct ambiguity sets can be found in Pflug and Wozabal
(2007).
Given the above definition of the ambiguity set and κ > 0, we arrive at the robustified
8
problem, the robust counterpart of (2):
infw∈RN supQ∈Bκ(P) R(⟨XQ,w⟩)
s.t. ⟨w,1⟩ = 1.(4)
The parameter κ signifies the degree of ambiguity, i.e. the uncertainty about the probability
model P. In problem (4), the decision maker deals with the ambiguity by adopting a worst case
approach, i.e. choosing the portfolio weights in such a way that the resulting decision is robust
with respect to the model uncertainty present in the problem.
If κ = 0, the problem reduces to the minimization of R(⟨XP,w⟩) in w, i.e. the nominal
problem (2). On the other hand, if κ increases, the decision will become more conservative as
the supremum in (4) is taken over a growing set of measures. It seems plausible to conjecture
that as κ → ∞, the weight of the information, represented by the measure P, diminishes and
the optimal decisions tend to a more diversified portfolio, approaching the uniform investment
strategy wu = (N−1, . . . ,N−1)⊤ in the limit.
Purpose of this paper is to prove the correctness of this conjecture for a large class of risk
functionals, which includes the examples mentioned at the beginning of this section.
3. Uniform investment strategy as a robust risk minimizing strategy
We will focus on convex, version independent risk functionals R : Lp(Ω, σ, µ) → R with
p < ∞, which admit a dual characterization of the form
R(X) = max E(XZ) − R(Z) : Z ∈ Lq (5)
where q is such that 1p +
1q = 1 and R : Lq(Ω, σ, µ)→ R is convex. Note that if R is lower semi-
continuous, then it admits a representation of the form (5), with R = R∗ where R∗ is the convex
conjugate of R. However, we do not require R = R∗ for the purpose of this paper; see Pflug and
Romisch (2007) for a discussion. We call a risk measure version independent or law invariant
if R(X1) = R(X2) for all random variables X1 and X2 which have the same distribution. Note
that if R = R∗ and X is in the interior of the domain X ∈ Lp(Ω, σ, µ) : R(X) < ∞ , then
argmaxZ E(XZ) − R(Z) = ∂R(X)
9
where ∂R(X) is the set of subgradients of R at X. For ease of notation, we will, therefore,
denote the set of maximizers of (5) at X by ∂R(X), even though ∂R(X) does not have to be the
set of subgradients for the case R , R∗.
We start by proving the following Lemma, which investigates how much the riskiness of a
decision w can change with changes in the distributions of the losses.
Lemma 1. Let 1 ≤ p < ∞ and R : Lp(Ω, σ, µ) → R be a convex, version independent risk
measure with dual representation (5). Let further q > 1 be such that 1p +
1q = 1 and w ∈ RN ,
then
|R(⟨XP1 ,w⟩) − R(⟨XP2 ,w⟩)| ≤ supZ:R(Z)<∞
||Z||Lq ||w||q dp(P1, P2) (6)
for arbitrary measures P1 and P2 on RN .
Proof. Let π be the optimal transport plan between P1 and P2 and choose Y : (Ω, σ, µ) →
RN × RN such that the image measure of Y on RN × RN is π. Call the projections of Y on
the first and the second component XP1 and XP2 respectively. Note that, as suggested by the
notation, the image measure of XPi is Pi, i = 1, 2.
Now choose a Z as a maximizer of (5) at the point ⟨XP1 ,w⟩, then
R(⟨XP1 ,w⟩) − R(⟨XP2 ,w⟩) ≤ E(⟨XP1 ,w⟩Z) − R(Z) − E(⟨XP2 ,w⟩Z) + R(Z)
≤ ||Z||Lq
(∫Ω
|⟨XP1 − XP2 ,w⟩|pdµ) 1
p
≤ ||Z||Lq ||w||q∫Ω
N∑n=1
|XP1n − XP2
n |pdµ
1p
= ||Z||Lq ||w||q∫
Rn×Rn
N∑n=1
|x1n − x2
n|pdπ(x1, x2)
1p
= ||Z||Lq ||w||qdp(P1, P2)
where the second and third step follow from Holders inequality, while the last two follow from
the definition of the variables XP1 , XP2 , and π. The result finally follows by repeating the
argument for R(⟨XP2 ,w⟩) − R(⟨XP1 ,w⟩) and taking the supremum over all the Z.
Obviously, the statement only makes sense if the upper bounds are finite. In this case,
the Lipschitz continuity of a class of risk measures with respect to the Kantorovich metric is
10
established. Since
wu = argminw∈RN :⟨w,1⟩=1 ||w||q, for all q ≥ 1, (7)
inspecting the right hand side in (6), we see that the bound is the smallest for w = wu. Hence,
showing that the bound is always achieved would establish that the difference in risk for dif-
ferent measures is always the smallest for the uniform investment strategy. To show that this is
indeed the case, we fix P and a radius κ > 0 and construct a measure Q for which dp(P,Q) = κ
and (6) holds with equality. We formalize this in the next Proposition.
Proposition 1. Let R : Lp(Ω, σ, µ) → R be a convex, version independent risk measure as in
Lemma 1 and let 1 < p < ∞ and q be defined by 1p +
1q = 1. Let further P be a probability
measure on RN and assume that
||Z||Lq = C for all Z ∈∪X∈Lp
∂R(X) with R(Z) < ∞. (8)
Then it holds that for every κ > 0 and every w ∈ RN , there is a measure Q on RN such that
dp(P,Q) = κ and
|R(⟨XQ,w⟩) − R(⟨XP,w⟩)| = Cκ||w||q,
i.e. the bound of Lemma 1 holds with equality.
Proof. Fix a Z ∈ ∂R(⟨XP,w⟩) with R(Z) < ∞ and define a random variable XQ = (XQ1 , . . . , X
QN )
by setting XQn = XP
n + c1(n)|wn|qp with
c1(n) =sign(wn) sign(Z)c2
||w||qq|Z|
qp
for all n : 1 ≤ n ≤ N and c2 > 0. If we set c1 = |c1(n)|, it is easily verified that
cp1 |wn|q = |XQ
n − XPn |p, ∀n : 1 ≤ n ≤ N (9)
holds. Furthermore, we have∣∣∣∣∣∣∣N∑
i=1
wn(XQn − XP
n )
∣∣∣∣∣∣∣p
=
∣∣∣∣∣∣∣N∑
n=1
wnc1(n)|wn|qp
∣∣∣∣∣∣∣p
= cp1
∣∣∣∣∣∣∣N∑
n=1
|wn|q∣∣∣∣∣∣∣p
= cp1 ||w||
pqq = cp
2 |Z|q. (10)
Note that the choice of the parameter c2 > 0 determines the distance dp(P,Q) of the image
measure Q of XQ to P, i.e. bigger values yield a bigger distance, and for every κ > 0, there is
11
a c2 > 0 such that dp(P,Q) = κ for the respective image measure Q. Assume that c2 is chosen
like that, then
R(⟨XQ,w⟩) − R(⟨XP,w⟩) ≥ E(⟨XQ,w⟩Z) − R(Z) − E(⟨XP,w⟩Z) + R(Z) (11)
= E(⟨XQ − XP,w⟩Z)
= ||Z||Lq
(∫Ω
|⟨XQ − XP,w⟩|pdµ) 1
p
(12)
= ||Z||Lq ||w||q∫Ω
N∑n=1
|XQn − XP
n |pdµ
1p
(13)
≥ ||Z||Lq ||w||qκ ≥ 0
where inequality (11) follows from the choice of Z. Equality in (12) follows from (10) and
sign(Z) = sign(⟨XQ − XP,w⟩),
which in turn is a consequence of the choice of c1. Finally, (13) follows from (9) and last
inequality by the definition of the Kantorovich distance. The assumptions on the subgradients,
together with Lemma 1, yield the desired result with Q the image measure of XQ.
Note that it follows from (13) and Lemma 1, that
dp(P,Q) =
∫Ω
N∑n=1
|XQn − XP
n |pdµ
1p
(14)
for the worst case measure Q defined in the proof of Proposition 1.
Although slightly different, the case p = 1 can be handled in a similar fashion.
Proposition 2. Let R : L1(Ω, σ, µ) → R be a convex, version independent risk measure like in
Lemma 1. Assume that
||Z||L∞ = C and |Z| = C or |Z| = 0 (15)
almost everywhere for all possible subgradients of R. Then it holds that for every probability
measure P on RN and κ > 0, there is a measure Q on RN such that d1(P,Q) = κ and
|R(⟨XP,w⟩) − R(⟨XQ,w⟩)| = C||w||∞κ,
i.e. the bound of Lemma 1 holds with equality.
12
Proof. The proof proceeds along the same lines as the proof of Proposition 1, with the only
difference that the definition of XQ = (XQ1 , . . . , X
QN ) changes to XQ
n = XPn+c1(n) for n : 1 ≤ n ≤ N
with
c1(n) =
sign(wn) sign(Z)c2, |wn| = ||w||∞
0, otherwise,(16)
where we define sign(0) = 0.
The conditions (8) and (15) on the subgradients in Propositions 1 and 2 might seem restric-
tive at the first glance. However, the conditions in Propositions 1 and 2 are valid for most of
the common risk measures. Two important examples are given below.
Example 1 (Conditional Value-at-Risk). The dual representation of CVaR is given by
CVaRα(X) = sup
E(XZ) : E(Z) = 1, 0 ≤ Z ≤ 11 − α
for 0 < α ≤ 1 (see Pflug and Romisch, 2007). We apply Proposition 2, since the CVaR is
defined on L1(Ω, σ, µ). If we choose a set A ⊆ Ω such that µ(A) = 1− α and X(ω) ≥ F−1X (α) for
all ω ∈ A, then it is easy to see that
Z(ω) =
1
1−α , ω ∈ A
0, otherwise∈ ∂CVaRα(X).
Hence, condition (15) of Proposition 2 is fulfilled.
Example 2 (Markowitz Functional). The natural domain of the Markowitz functional is L2(Ω, σ, µ).
To derive it’s dual formulation, note that√Var(X) = ||X − E(X)||L2 = sup E((X − E(X))Z) : ||Z||L2 = 1
= sup E(X(Z − E(Z)) : ||Z||L2 ≤ 1
= sup E(XZ) : E(Z) = 0, ||Z||L2 = 1 .
Therefore, we obtain
Mγ(X) = E(X) + γ sup E(XZ) : E(Z) = 0, ||Z||L2 = 1
= sup E(X(γZ + 1)) : E(Z) = 0, ||Z||L2 = 1
= supE(XZ) : E(Z) = 1, ||Z||L2 =
√1 + γ2
.
13
Hence, it is immediate that assumption (8) in Proposition 1 is fulfilled.
Proposition 1 and 2 show that, for given portfolio weights w,
supQ∈Bκ(P)
R(⟨XQ,w⟩) = R(⟨XP,w⟩) +C||w||qκ. (17)
The solution Q of (17) can be found as the image measure of XQ. By (7), given the budget
constraint ⟨w,1⟩ = 1, the smallest change occurs for the uniform portfolio wu. To find w∗
which solves (4) for a given κ > 0, we have to consider the tradeoff between choosing a
portfolio which fares well under the original measure P and the robustness of the choice with
respect to the ambiguity. However, it can be immediately seen that for every admissible w there
is a level κ, such that
R(⟨XP,wu⟩) +C||wu||qκ < R(⟨XP,w⟩) +C||w||qκ.
Hence, as κ → ∞, the optimal portfolio converges to wu. We formalize this finding in the next
Proposition.
Proposition 3. Let 1 ≤ p < ∞ and R be a convex risk measure as in Proposition 1 or Proposi-
tion 2, then, as κ → ∞, in problem (4), the optimal portfolios converge to the uniform portfolio
wu. More specifically:
1. If p = 1 then wu is the optimal solution to problem (4) for κ > κ∗ with
κ∗ = (N − 1)E(||XP||11Z,0
).
2. If p = 2, then the optimal portfolio w∗ solving (4) satisfies ||w∗ − wu||2 ≤ D, if
κ ≥( 1
ND2 + 1) 1
2
+1√
ND
E(||XP||221Z,0
) 12.
3. If p < 1, 2, then for every ϵ > 0, there is a κϵ such that for κ > κϵ the optimal solution
w∗ for (4) fulfills ||w∗ − wu||q < ϵ.
Proof. We start by stating the following inequality
R(⟨XP,w1⟩) − R(⟨XP,w2⟩) ≤ C||w1 − w2||qE(||XP||pp1Z,0
) 1p (18)
14
for all Z ∈ ∂R(⟨XP,w1⟩). (18) can be proven using a similar argument as employed in Lemma
1. By (17), the uniform portfolio is optimal for problem (4) among a given set of portfolios B,
iff
R(⟨XP,wu⟩) +C||wu||qκ ≤ R(⟨XP,w⟩) +C||w||qκ, ∀w ∈ B
which, using (18), is implied by
κ ≥||w − wu||q||w||q − ||wu||q
E(||XP||pp1Z,0
) 1p, ∀w ∈ B. (19)
For the case p = 1, let n∗ = argmax1≤n≤N |wn − 1/N|. If wn∗ > 1/N, then ||w − wu||∞ =
||w||∞ − ||wu||∞. If, on the other hand, wn∗ < 1/N, then wn∗ = minn wn and we conclude that
maxn
wn ≥1N+
1/N − wn∗
N − 1.
It follows that
(N − 1)(||w||∞ − 1/N) ≥ 1/N − wn∗ = ||w − wu||∞
establishing the first part of the Proposition.
For p = q = 2, let f2, . . . , fN orthogonal to each other and to wu with || fi||2 = 1 for i =
2, . . . ,N. Hence, any w with ⟨w,1⟩ = 1 can be written as w = wu+∑N
i=2 ci fi with c2, . . . , cN ∈ R
and
||w − wu||2||w||2 − ||wu||2
=||w − wu||2(
1/N +∑N
i=2 c2i
) 12 − 1/
√N=
||w − wu||2(1/N + ||w − wu||22
) 12 − 1/
√N
=
(1
N||w − wu||22+ 1
) 12
+1
√N||w − wu||2
.
Clearly, as ||w − wu||2 → ∞, the above expression tends to 1, while it approaches ∞ for ||w −
wu||2 → 0. Hence, it follows that
||w − wu||2||w||2 − ||wu||2
≤(
1ND2 + 1
) 12
+1√
ND, ∀w : ||w − wu||2 ≥ D.
This, together with (19), establishes the second statement.
For p < 1, 2, let (xn)n∈N be a sequence with xn ∞ and define the convex sets
An =w ∈ RN : ⟨w,1⟩ = 1, R(⟨XP,wu⟩) +C||wu||qxn ≥ R(⟨XP,w⟩) +C||w||qxn
15
in RN . An+1 ⊆ An for all n ∈ N and∞∩
n=1
An = wu .
Since (2) is well-posed, the mapping w 7→ R(⟨XP,w⟩)+C||w||qxn is inf-compact, i.e. the sets An
are compact. For ϵ > 0, define the compact sets Bϵn = An \w ∈ RN : ||w − wu|| < ϵ
, and note
that by the above∞∩
n=1
Bϵn = ∅
and by compactness, there is a Mϵ ∈ N such that∩Mϵ
n=1 Bϵn = ∅. Hence, we have shown that for
every ϵ > 0, there is a Mϵ ∈ N such that the optimal solution w∗ for (4) fulfills ||w∗ − wu||q < ϵ
for κ > xMϵ . Setting κϵ = xMϵ concludes the proof.
4. Numerical study
In this section, we will demonstrate the results of the previous section using real market
data. In particular, we solve problem (4) for the Markowitz functional, and the Conditional
Value-at-Risk and investigate the optimal portfolios as the degree of ambiguity increases. As a
byproduct, we derive robust counterparts of the two risk functionals, which lead to ambiguous
optimization problems of the same computational complexity as the nominal problems with the
original measures. We demonstrate that for p = 1, the threshold κ, for the uniform portfolio
to be optimal, is actually smaller than the bound in Proposition 3. Similarly, we show that
||w∗ − wu||2 is actually smaller than the bound derived in Proposition 3 for the case p = 2.
The asset universe for the numerical study consists of the following seven indices: the
Dow Jones Industrial index (DJI), the Dow Jones CBOT Treasury Index (CBTI), SPDR Gold
Shares (GLD), the Dow Jones Composite All REIT (RCIT), the Euro Stoxx 50 (STOXX50),
the Nikkei 225 index (N225), and the Shanghei Stock Exchange Composite Index (SSEC). The
assets are all quoted in US dollars, i.e. the assets that are originally quoted in another currency
are multiplied with the respective exchange rates.
We use historical weekly return data for the period 01.01.2007 until 31.10.2010 to obtain
scenarios for the joint asset returns. In all we use 151 data points, each of which we assign
the same probability, i.e. the measure P equals the empirical measure constructed from these
16
151 historical asset returns. While in Section 4.1 the scenarios are used directly, facilitating a
scenario based approach to robustified Conditional Value-at-Risk optimization, in Section 4.2,
they are used to estimate the expected return as well as the covariance matrix needed for the
robustified Markowitz approach.
4.1. Conditional Value-at-Risk
We start our investigation by defining the Ambiguous Conditional Value-at-Risk as
A-CVaRα(⟨w, XP⟩, κ) = maxQ∈Bκ(P)
CVaRα(⟨w, XQ⟩)
and consider the problem
minw A-CVaRα(⟨w, XP⟩, κ)
s.t. ⟨w,1⟩ = 1.(20)
To ensure that the worst case distribution Q is exactly at distance κ from P, we use (14) and
(16) and choose
c2 =1
(1 − α)kκ
where k = | n : |wn| = ||w||∞ |. Therefore, by (16), we have that XQ = (XQ1 , . . . , X
QN )⊤ with
XQn =
XP
n + sign(wn) sign(Z) κ(1−α)k , |wn| = ||w||∞
XPn , otherwise
for n : 1 ≤ n ≤ N and Z ∈ ∂CVaRα(⟨XP,w⟩). Furthermore,
⟨w, XQ⟩ = ⟨w, XP⟩ + 1Z,0||w||∞κ
1 − α.
In a finite scenario setting with loss scenarios x1, . . . , xS and probabilities p1, . . . , pS under
the measure P, problem (20) can be cast as the following linear programming problem
infw∈RN ,M∈R a + 11−α
∑Ss=1 zs ps
s.t. zs ≥ ⟨w, xs⟩ + M − a, ∀s ∈ 1, . . . , S
⟨w,1⟩ = 1
wnκ
1−α ≤ M, ∀n ∈ 1, . . . ,N
zs ≥ 0, ∀s ∈ 1, . . . , S .
17
0 0.005 0.01 0.015 0.02 0.025 0.030
0.2
0.4
0.6
0.8
1
Radius
Com
posi
tion
CBTI DJI GLD N225 RCIT SSEC STOXX50
(a)0 0.005 0.01 0.015 0.02 0.025 0.03
0
0.1
0.2
0.3
0.4
0.5
0.6
Radius
Her
finda
hl−
Hirs
chm
an In
dex
(b)
Figure 1: In (a) the optimal portfolios in dependence of κ are depicted. (b) shows the corresponding values for the
Herfindahl-Hirschman index.
The portfolio compositions for different levels of κ are depicted in Figure 1a. Every vertical cut
in the picture represents the portfolio composition for a given level of κ. For small values of κ,
some weights are negative which results in overall investment larger than one. It can be seen
that as κ increases, the portfolios rapidly approach the uniform portfolio. This observation is
supported by Figure 1b, which depicts the normalized Herfindahl-Hirschman index values for
the portfolios. Recall that this index is defined as [∑N
n=1 w2n − 1/N]/[1− 1/N]. It takes the value
0 for the uniform strategy wu and the value 1 for the investment in just one asset.
In the above example, the lowest level of κ, for which the optimal decision is wu, is 0.026,
and the analytical bound κ∗ from Proposition 3 is equal to 0.0734.
4.2. Markowitz functional
Analyzing the derivation of the dual representation for the Markowitz functional, we deduce
that for a given X, the subgradient Z at Mγ(X) is given by
Z = γX − E(X)||X − E(X)||2
+ 1.
18
To construct a worst case measure with d2(P,Q) = κ, we use (14) and note that
d2(P,Q) =
∫Ω
N∑n=1
|XQn − XP
n |2dµ
12
=
∫Ω
N∑i=1
∣∣∣∣∣∣sign(Z) sign(wn)c2
||w||22|wn|Z
∣∣∣∣∣∣2 dµ
12
=
∫Ω
c22
||w||42|Z|2
N∑i=1
|wn|2dµ
12
=c2
||w||2
(∫Ω
|Z|2dµ) 1
2
=c2
||w||2||Z||L2
and therefore, c2 =κ||w||2√
1+γ2for a given κ > 0.
We proceed by deriving a representation of the Ambiguous-Markowitz-Functional
A-Mγ(⟨XP,w⟩, κ) = maxQ∈Bκ(P)
Mγ(⟨w, XQ⟩).
By (17), the worst case equivalent of the Markowitz risk measure is
A-Mγ(⟨XP,w⟩, κ) = Mγ(⟨XP,w⟩) + κ||w||2√
1 + γ2.
Solving the problem
minw A-Mγ(⟨w, XP⟩, κ)
s.t. ⟨w,1⟩ = 1(21)
numerically, we obtain the portfolio weights depicted in Figure 2a; Figure 2b shows the corre-
sponding values of the Herfindahl-Hirschman index. The results confirm the theoretical find-
ings of Proposition 3. The optimal portfolios converge to wu, but there seems to be no finite
value of κ such that the optimal portfolios are actually equal to wu. Nevertheless, as is evident
from Figure 2, convergence is rather fast, and even for small values of κ, the optimal portfolios
are very close to wu. In this sense, the convergence is faster than for the CVaR case. Figure 3
depicts the actual distance of the optimal portfolios to wu as well as the theoretical bound. In
line with the results on CVaR and the discussion above, the plot shows that the actual distance
turns out to be much smaller than the theoretical bound.
19
0 0.05 0.1 0.15 0.2 0.25 0.30
0.2
0.4
0.6
0.8
1
Radius
Com
posi
tion
CBTI DJI GLD N225 RCIT SSEC STOXX50
(a)0 0.05 0.1 0.15 0.2 0.25 0.3
0.1
0.2
0.3
0.4
0.5
0.6
Radius
Her
finda
hl−
Hirs
chm
an In
dex
(b)
Figure 2: In (a) the optimal portfolios in dependence of κ are depicted. (b) shows the corresponding values for the
Herfindahl-Hirschman index.
0 0.05 0.1 0.15 0.2 0.25 0.3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Radius
Dis
tanc
e to
1/N
Str
ateg
y
Figure 3: Actual distance of the optimal portfolios w∗ to wu measured in the 2-norm (solid line) versus theoretical
bound (dotted line).
20
5. Conclusion
We showed that the uniform investment strategy or 1/N rule is a rational strategy to follow
in stochastic portfolio decision problems where the distribution of asset returns is ambiguous,
and the decision maker adopts a worst case approach taking into account all measures in an
ambiguity set. The ambiguity set consists of all measures in a neighborhood of a reference
measure, which represents the prior information of the decision maker. We use the Kantorovich
metric to construct the ambiguity sets around the reference measure in a non-parametric way,
i.e. we do not impose any restrictions on the measures. The choice of the Kantorovich metric is
natural since it allows the construction of ambiguity sets using statistical tools and furthermore
is closely related to existing stability theory for stochastic programming problems.
In the second part of the paper, we numerically demonstrate the convergence to the uniform
portfolio in portfolio optimization problems with the Markowitz functional and Conditional
Value-at-Risk as the objective function. The results show that the optimal portfolio converges
to the uniform portfolio even faster than suggested by the theoretical bounds established in
Section 3. Furthermore, we show how the structure of the portfolio actually approaches the
uniform portfolio, i.e. how even small levels of ambiguity cause diversification in the optimal
portfolios. This point is illustrated by the fact that the normalized Herfindahl-Hirschman index
of the portfolios is monotonically decreasing with the degree of ambiguity in the model.
The results obtained in this paper contribute to the contemporary discussion in two ways:
1. We showed that a rational agent chooses increasingly diversified portfolios, when model
uncertainty increases. This may serve as an explanation for the empirically observed use
of simple diversification heuristics in portfolio selection settings. The paper therefore
provides a justification of this behavioral pattern founded in the theory of rational choice.
2. The optimality of the uniform portfolio rule in the face of model uncertainty explains
the good performance of this strategy in comparative studies, such as DeMiguel et al.
(2009b). If naive diversification outperforms more sophisticated models, this can be seen
as a clear indication that the modeling of the data generating process is not accurate
enough to serve as an input for the particular model class. This, in turn, implies that the
decision maker either has to improve on the statistical modeling, or if this is not possible,
21
choose a different criterion of optimality which is more robust with respect to estimation
error. The bounds derived in Proposition 3 may serve as an indication of the sensitivity
of different risk measures to model uncertainty.
Further research on the topic could reveal a more systematic characterization of the different
risk measures with respect to model uncertainty.
Appendix A. Random variables with given image measures
Lemma 2. Let ([0, 1], σ[0,1], λ) be the standard probability space with the Lebesgue measure on
the Borel sets σ[0,1]. Let further M be a complete, separable, uncountable metric space and P a
Borel probability measure on M. Then there is a measurable function XP : ([0, 1], σ[0,1], λ) →
(M, σM, P) such that
P(A) = λ((XP)−1(A)), ∀A ∈ σM.
Proof. Let K ∈ N ∪ 0,∞ and x1, . . ., xK be atoms of P with probabilities p1, . . ., pK (if P has
no atoms then set K = 0) and p =∑K
k=1 pk. Define A1 = [0, p1) and
Ak =
k−1∑j=1
p j,
k∑j=1
p j
for k = 2, . . ., K, and note that λ(Ak) = pk. Define the measure P′ by
P′(A) = P(A) −∑
xk:xk∈AP(xk)
and note that under the conditions of the Lemma there is a measure preserving map T :
([ p, 1], σ[p,1], λ)→ (M, σM, P′) by Theorem 15.5.16 in Royden (1988). Defining
XP(x) =
xk, x ∈ Ak, k = 1, . . . ,K
T (x), x ∈ M \∪Kk=1 Ak
concludes the proof.
22
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